Intelligent identification method of sludge bulking based on type-2 fuzzy neural network

ABSTRACT

An intelligent identification method of sludge bulking based on type-2 fuzzy-neural-network belongs to the field of intelligent detection technology. The sludge volume index (SVI) in wastewater treatment plant is an important index to measure the sludge bulking of activated sludge process. However, poor production conditions and serious random interference in sewage treatment process are characterized by strong coupling, large time-varying and serious hysteresis, which makes the detection of SVI concentration of sludge volume index extremely difficult. At the same time, there are many types of sludge bulking faults, which are difficult to identify effectively. Due to the sludge volume index (SVI) is unable to online monitoring and the fault type of sludge bulking is difficult to determined, the invention develop soft-computing model based on type-2 fuzzy-neural-network to complete the real-time detection of sludge volume index (SVI). Combined with the target-related identification algorithm, the fault type of sludge bulking is determined. Results show that the intelligent identification method can quickly obtain the sludge volume index (SVI), accurate identification fault type of sludge bulking, improve the quality and ensure the safety operation of the wastewater treatment process.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the priority benefit of Chinese applicationserial no. 201810790763.9, filed Jul. 18, 2018. All disclosure of theChina application is incorporated herein by reference.

TECHNOLOGY AREA

Based on the running characteristics of wastewater treatment process andusing type-2 fuzzy neural network, the invention design an intelligentidentification method to realize the real-time measurement about indexSVI of sludge bulking in wastewater treatment process and theidentification for fault categories of sludge bulking, where theconcentration of sludge volume index SVI in wastewater treatment plantis the performance index of measuring the coagulation sedimentation andthickening properties of activated sludge. The prediction of sludgevolume index SVI and identification of fault category of sludge bulkinghave a great significance for realizing the monitoring and controllingin wastewater treatment process. It have great important influence forenergy saving and safety operation to apply intelligent recognitionmethod in wastewater treatment system. Moreover, the above methodsbelonging to the control field and water treatment field simultaneously,is the important branch for the field of advanced manufacturingtechnology. Therefore, the intelligent recognition of the sludge bulkingis of great significance in the wastewater treatment system.

TECHNOLOGY BACKGROUND

During the development in past hundred years, the wastewater treatmenttechnology of activated sludge process has become the mainstream ofurban wastewater treatment technology in the worldwide. However, inChina, the present wastewater treatment plant have the drawback thatbackward wastewater treatment equipment, low-end automation level andincomplete control system, which lead to the high frequency occurrenceof sludge bulking. Sludge bulking can lead to loose sludge structure,increasing sludge volume and reduction of sludge sedimentation velocityand then, there is difficult for solid settlement separation whichaffect water quality and damage the normal and stable operation ofactivated sludge system in the wastewater treatment process. Therefore,it is of great significance to analyze the phenomenon of sludge bulkingand study the diagnosis method of sludge bulking to improve theefficiency and ensure the normal operation of working condition forwastewater treatment process.

At present, many study for identifying sludge bulking phenomenon havedevelopment, but the implementation effect is not optimistic. On the onehand, due to the complex mechanism properties of sludge bulking,mechanism model based sludge bulking diagnosis method cannot cover allthe growth mechanism of microorganism to meets the stability and theaccuracy simultaneously. Moreover, mechanism model based sludge bulkingdiagnosis method judge sludge bulking by measuring the length, abundanceand others morphological characteristics of the microorganism what havethe characteristics of complex operation and strong time-delay and isdifficult to be applied in the wastewater treatment process. On theother hand, the dynamic nonlinear of wastewater treatment process maketraditional sludge bulking prediction model is difficult to adapt theworking condition with strong dynamic change and identify the fault typeof sludge bulking accurately. Based on the negative influence of thesludge bulking, design a sludge bulking diagnostic method whichcharacter real-time dynamic tracking, accurate and stable have importanttheoretical significance and application value for monitoring wastewatertreatment process running online, stabilizing the wastewater treatmentprocess, preventing the sludge bulking phenomenon, improving the waterquality and increasing the efficiency of wastewater treatment.

The invention proposed a type-2 fuzzy neural network based intelligentdiagnosis method about sludge bulking. The type-2 fuzzy neural networkmodel and parameter optimization algorithm can improve the predictionperformance of the network effectively and then, the fault typeidentification of sludge bulking is realized by target correlationrecognition algorithm. The above intelligent identification method canrealize the real-time detection of sludge volume index SVI andidentification of fault type for sludge bulking. Moreover, theintelligent identification method also reduce the measurement cost,increase the identification accuracy, provides a fast and efficientidentification mean and improve the benefit of the wastewater treatmentplant.

SUMMARY OF THE PATENT

The invention proposed a type-2 fuzzy neural network based sludgebulking intelligent diagnosis method. First, the method analyze thewastewater treatment process and select a group of auxiliary variablewhich is closely related to the sludge volume index SVI and easilymeasured. Then, type-2 fuzzy neural network is used for measuring thesludge volume index SVI online. In the end, the target correlationrecognition algorithm is applied for identifying the fault type ofsludge bulking. The method solves the problem about long measurementcycle of sludge volume index SVI and identify difficulty of sludgebulking category.

The invention includes the following steps:

(1) Determine the input and output variables of sludge volume index(SVI): in the activated sludge wastewater treatment process, the inputvariables of SVI soft-computing model include: dissolved oxygen (DO)concentration, total nitrogen (TN) concentration, organic load rate(F/M), pH, T. The output value of soft-computing model is the SVIvalues. The sludge bulking contains the following fault types: low DOconcentration, nutrient deficit, low sludge loading, low pH, and lowtemperature;

(2) SVI soft-computing model: establish SVI soft-computing model basedon type-2 fuzzy-neural-network, the structure of type-2fuzzy-neural-network contains five layers: input layer, membershipfunction layer, firing layer, consequent layer and output layer, thenetwork is 5-M-L-2-1, including 5 neurons in input layer, M neurons inmembership function layer, L neurons in firing layer, 2 neurons inconsequent layer and 1 neurons in output layer, M and L are the integerslarger than 2; connecting weights between input layer and membershipfunction layer are assigned 1; the number of the training sample is N,the input of type-2 fuzzy-neural-network is x(t)=[x₁(t), x₂(t), x₃(t),x₄(t), x₅(t)] at time t, x₁(t) represents the DO concentration at timet; x₂(t) represents the TN concentration at time t, x₃(t) represents theorganic load rate (F/M) value at time t, x₄(t) represents the pH valueat time t, and x₅(t) represents the T value at time t, the output oftype-2 fuzzy-neural-network is y_(d)(t) and the actual output is y(t);type-2 fuzzy-neural-network includes:

{circle around (1)} input layer: there are 5 neurons in this layer, theoutput is:o _(i)(t)=x _(i)(t)  (1)where o_(i)(t) is the ith output value at time t, i=1, 2, . . . , 5,

{circle around (2)} membership function layer: there are M neurons inmembership function layer, the output is:

$\begin{matrix}{{{\tau_{m}^{i}(t)} = {{N\left( {{c_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)} = {\exp\left\{ {{- \frac{1}{2}}\left( \frac{{o_{i}(t)} - {c_{m}^{i}(t)}}{\sigma_{m}^{i}(t)} \right)^{2}} \right\}}}},{i = 1},2,\ldots\;,{5;{m = 1}},2,\ldots\;,M,} & (2) \\{{{c_{m}^{i}(t)} \in \left\lbrack {{{\underset{\_}{c}}_{m}^{i}(t)},{{\overset{\_}{c}}_{m}^{i}(t)}} \right\rbrack},} & (3)\end{matrix}$where τ^(i) _(m)(t) is the mth membership function with the ith input attime t, N represents the membership function, c^(i) _(m)(t) is theuncertain center of the mth membership function neuron with the ithinput at time t, c^(i) _(m)(t) is the lower center value of the mthmembership function neuron with the ith input at time t, c ^(i) _(m)(t)is the upper center value of the mth membership function neuron with theith input at time t (where the initial lower center value and initialupper center value of the mth membership function neuron with the ithinput i.e., c ^(i) _(m)(0) and c^(i) _(m)(0) is obtained by that randominitial center of the mth membership function neuron with the ith inputc^(i) _(m)(0) add and subtract a constant), σ^(i) _(m)(t) is thestandard deviation of the mth membership function neuron with the ithinput at time t, the bounded internal of τ^(i) _(m)(t) is [τ ^(i)_(m)(t), τ ^(i) _(m)(t)]

$\begin{matrix}{{{\underset{\_}{\tau}}_{m}^{i}\left( {o_{i}(t)} \right)} = \left\{ {\begin{matrix}{{N\left( {{{\overset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} \leq {\left( {{{\underset{\_}{c}}_{m}^{i}(t)} + {{\overset{\_}{c}}_{m}^{i}(t)}} \right)/2}} \\{{N\left( {{{\underset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} > {\left( {{{\underset{\_}{c}}_{m}^{i}(t)} + {{\overset{\_}{c}}_{m}^{i}(t)}} \right)/2}}\end{matrix},} \right.} & (4) \\{{{\overset{\_}{\tau}}_{m}^{i}\left( {o_{i}(t)} \right)} = \left\{ {\begin{matrix}{{N\left( {{{\underset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} \leq {{\underset{\_}{c}}_{m}^{i}(t)}} \\{1,} & {{{\underset{\_}{c}}_{m}^{i}(t)} < {o_{i}(t)} < {{\overset{\_}{c}}_{m}^{i}(t)}} \\{{N\left( {{{\overset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} > {{\overset{\_}{c}}_{m}^{i}(t)}}\end{matrix},} \right.} & (5)\end{matrix}$where τ ^(i) _(m)(t) and τ ^(i) _(m)(t) are the lower value and uppervalue of the mth membership function neuron with the ith input at timet, respectively, 0<τ ^(i) _(m)(t)≤τ ^(i) _(m)(t)≤1,

{circle around (3)} firing layer: there are L neurons in this layer, andthe output values are:

$\begin{matrix}{{{F_{l}(t)} = \left\lbrack {{{\underset{\_}{f}}_{l}(t)},{{\overset{\_}{f}}_{l}(t)}} \right\rbrack},{{{\underset{\_}{f}}_{l}(t)} = {\prod\limits_{i = 1}^{5}\;{{\underset{\_}{\tau}}_{m}^{i}(t)}}},{{{\overset{\_}{f}}_{l}(t)} = {\prod\limits_{i = 1}^{5}\;{{\overset{\_}{\tau}}_{m}^{i}(t)}}},{l = 1},2,\ldots\;,L,} & (6)\end{matrix}$where F_(l)(t) is the firing strength of the lth firing neuron, f_(l)(t) and f _(l)(t) are the lower firing strength and upper firingstrength of the lth firing neuron, respectively, 0<f _(l)t)≤f _(l)(t)≤1,

{circle around (4)} consequent layer: this layer contains two consequentneurons, the output values are

$\begin{matrix}{{{\underset{\_}{y}(t)} = \frac{\sum\limits_{l = 1}^{L}{{{\underset{\_}{f}}_{l}(t)}{a_{l}(t)}}}{\sum\limits_{l = 1}^{L}{{\underset{\_}{f}}_{l}(t)}}},{{\overset{\_}{y}(t)} = \frac{\sum\limits_{l = 1}^{L}{{{\underset{\_}{f}}_{l}(t)}{a_{l}(t)}}}{\sum\limits_{l = 1}^{L}{{\underset{\_}{f}}_{l}(t)}}},{{a_{l}(t)} = {\sum\limits_{i = 1}^{5}{{\theta_{l}^{i}(t)}{x_{i}(t)}}}},} & (7)\end{matrix}$where y(t) and y(t) are the low and up output values of the consequentneurons, a_(l)(t) is weight of the lth firing neuron, θ^(i) _(l)(t) isthe weight coefficient of the lth firing neuron with the ith input attime t,

{circle around (5)} output layer: the output value is:y(t)=η(t) y (t)+(1−η(t)) y (t)  (8)where η(t) and y(t) are the proportion of the low output and the outputvalue of type-2 fuzzy-neural-network, the error of type-2fuzzy-neural-network is:

$\begin{matrix}{{E(t)} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\left( {{y_{d}(t)} - {y(t)}} \right)^{2}}}} & (9)\end{matrix}$where y_(d)(t) is the output of type-2 fuzzy-neural-network and theactual output is expressed as y(t);

(3) train type-2 fuzzy-neural-network

{circle around (1)} give the type-2 fuzzy-neural-network, the initialnumber of firing layer neurons is M, M>2 is a positive integer; theinput of type-2 fuzzy-neural-network is x(1), x(2), . . . , x(t), . . ., x(N), correspondingly, the output is y_(d)(1), y_(d)(2), . . . ,y_(d)(t), . . . , y_(d)(N), expected error value is set to E_(d),E_(d)∈(0, 0.01),

{circle around (2)} set the learning step s=1;

{circle around (3)} t=s; according to Eqs. (1)-(7), calculate the outputof type-2 fuzzy-neural-network, exploiting adaptive second-orderalgorithm:ψ(t+1)=ψ(t)+(H(t)+λ(t)I)⁻¹ v(t)  (10)where ψ(t)=[c ^(i) _(m)(t), c^(i) _(m)(t), σ^(i) _(m)(t), η(t), w^(i)_(m)(t)] is the parameter matrix of type-2 fuzzy-neural-network at timet, c^(i) _(m)(t) is the lower center value of the mth membershipfunction neuron with the ith input at time t, c ^(i) _(m)(t) is theupper center value of the membership function neuron with the ith inputat time t, σ^(i) _(m)(t) is the standard deviation of the mth membershipfunction neuron with the ith input at time t, η(t) is the proportion ofthe lower output, θ^(i) _(l)(t) is the weight coefficient of the lthfiring neuron with the ith input at time t, H(t) is the quasi Hessianmatrix, v(t) is gradient vector, I is the identity matrix and λ(t) isthe adaptive learning rate defined as:λ(t)=γ|E(t)|+(1−γ)∥v(t)∥  (11)where γ∈(0, 1), the expression of H(t) and v(t) are defined as:H(t)=J ^(T)(t)J(t)  (12)v(t)=J ^(T)(t)E(t)  (13)where the Jacobian vector J(t) is calculated as:

$\begin{matrix}{{J(t)} = \left\lbrack {\frac{\partial{e(t)}}{\partial{{\underset{\_}{c}}_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{{\overset{\_}{c}}_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{\sigma_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{\eta(t)}},\frac{\partial{e(t)}}{\partial{\theta_{m}^{i}(t)}}} \right\rbrack} & (14)\end{matrix}$

{circle around (4)} according to Eq. (9), calculate the performance oftype-2 fuzzy-neural-network, if E(t)≥E_(d), go to step {circle around(3)}; if E(t)<E_(d), stop the training process;

(4) the target-related identification algorithm is used to determine thefault type of sludge bulking, which is specifically as follows:

{circle around (1)} the test samples is used as the input of the type-2fuzzy-neural-network, and the sludge volume index (SVI) is calculated;

{circle around (2)} if SVI≤150, it is determined that there is no sludgebulking during the wastewater treatment process;

{circle around (3)} if SVI>150, sludge bulking at the wastewatertreatment operation was determined and regression coefficients of allvariables were calculated:

$\begin{matrix}{{{b_{i}(t)} = \frac{{u_{i}(t)}^{T}{t_{i}(t)}}{{t_{i}(t)}^{T}{t_{i}(t)}}},} & (15)\end{matrix}$where b_(i)(t) is the regression coefficient of ith input at time t,b(t)=[b₁(t), . . . , b_(i)(t), . . . , b₅(t)] is the regressioncoefficient vector, u_(i)(t) is the ith score vector of the outputvector at time t, U(t)=[u₁(t), . . . , u_(i)(t), . . . , u₅(t)] is scorematrix of the output vector at time t, t_(i)(t) is the ith score vectorof the input matrix at time t, T(t)=[t₁(t), . . . , t_(i)(t), . . . ,t₅(t)] is score matrix of the input matrix at time t, u_(i)(t) and t_(i)(t) are given as:

$\begin{matrix}{{{u_{i}(t)} = \frac{{y(t)}{q_{i}(t)}}{{q_{i}(t)}^{T}{q_{i}(t)}}},} & (16) \\{{{t_{i}(t)} = \frac{{X(t)}{w_{i}(t)}}{{w_{i}(t)}^{T}{w_{i}(t)}}},} & (17)\end{matrix}$where q_(i)(t) is the ith loading value of output vector at time t,q(t)∈R^(1×5) is the loading vector of output vector at time t,y(t)=[y(t−K+1), y(t−K+2), . . . , y(t)]^(T), y(t) is the SVI value attime t, X(t)=[x₁(t), . . . , x_(i)(t), . . . , x₅(t)] is the inputmatrix of type-2 fuzzy-neural-network, x_(i)(t)=[x_(i)(t−K+1),x_(i)(t−K+2), . . . , x_(i)(t)]^(T), x_(i)(t) is the ith input variableat time t, w_(i)(t) is the ith feature vector at time t, W(t)=[w₁(t), .. . , w_(i)(t), . . . , w₅(t)] is the feature matrix of X(t)^(T)y(t),the expressions of q_(i)(t) and W(t) are

$\begin{matrix}{{{q_{i}(t)}^{T} = \frac{{t_{i}(t)}^{T}{y(t)}}{{t_{i}(t)}^{T}{t_{i}(t)}}},} & (18) \\{{{{W(t)}^{T}{\Lambda(t)}{W(t)}} = {E\left\{ {{X(t)}^{T}{y(t)}{y(t)}^{T}{X(t)}} \right\}}},} & (19)\end{matrix}$where Λ(t) is the eigenvalue matrices of X(t)^(T)y(t), The function Erepresents the eigenvector and eigenvalue of the matrix, the innerrelative model of y(t) and X(t) can be expressed as:

$\begin{matrix}\left\{ {\begin{matrix}{{X(t)} = {{{{T(t)}{P(t)}^{T}} + {\Delta(t)}} = {{\sum\limits_{i = 1}^{5}{{t_{i}(t)}{p_{i}(t)}^{T}}} + {\Delta(t)}}}} \\{{y(t)} = {{{{U(t)}{q(t)}^{T}} + {G(t)}} = {{\sum\limits_{i = 1}^{5}{{u_{i}(t)}{q_{i}(t)}^{T}}} + {G(t)}}}}\end{matrix},} \right. & (20)\end{matrix}$where Δ(t)∈R^(K×5) is the residual matrix of X(t), Δ(t)=[δ₁(t), . . . ,δ_(i)(t), . . . , δ₅(t)], where δ_(i)(t) present the residual vector ofith input. G(t)∈R^(K×1) is the residual vector of y(t);

{circle around (4)} when the regression coefficient of the inputvariable satisfies:b _(max)(t)=max b(t),  (21)where b_(max)(t) is the maximum regression coefficient of the inputvariables, and the corresponding fault type is the source of sludgebulking.The Novelties of this Patent Contain:

(1) Aiming at the problem that sludge bulking is difficult to identifyin current wastewater treatment process, the invention proposed a type-2fuzzy neural network based intelligent identification method. First,extract five relevant variables which is related to the sludge volumeindex SVI according to the work report of real-world wastewatertreatment plant: dissolved oxygen concentration DO, total nitrogen TN,load activated sludge, F/M power of hydrogen pH and temperature T andthen, realize the prediction of sludge volume index SVI. The inventionsolves the problem that sludge volume index SVI is hard to measureonline, avoid the application of complicated sensor and reduces therunning cost

(2) Based on target correlation identification method, the inventionidentify the category of sludge bulking through the contribution ofvariable to sludge bulking. The above method can not only measure theinfluence intensity of variable in the sludge bulking process, but alsoidentify the fault category of the sludge bulking what solve thedifficult identification of sludge bulking in the wastewater treatmentprocess. Moreover, the target correlation identification method is usedfor identifying the fault category of sludge bulking online, what hasthe characteristics of high precision and the strong adaptability toenvironmental variation.

Attention: the invention adopts type-2 fuzzy neural network and thetarget correlation identification algorithm to establish the sludgebulking intelligent recognition method. The research that adopt type-2fuzzy neural network and target correlation identification algorithm inthis invention for intelligent identification of sludge bulking, shouldfall within the scope of the present invention

DESCRIPTION OF DRAWINGS

FIG. 1 is the initial structural topology diagram of the type-2 fuzzyneural network.

FIG. 2 is the SVI test result diagram of the sludge volume index, wherethe blue line is the desired output value of sludge volume index SVI,and the black line is the predicted value of type-2 fuzzy neuralnetwork.

FIG. 3 is the SVI prediction error diagram of the sludge volume index.

FIG. 4 is the fault category identification diagram of the sludgebulking.

DETAILED DESCRIPTION OF THE INVENTION

The invention select the characteristic variables to measure sludgevolume index SVI as dissolved oxygen concentration DO, total nitrogenTN, load activated sludge F/M, power of hydrogen pH and temperature T,where pH no unit. The unit of temperature is Celsius. Others units aremg/l.

The experimental data come from the 2017 water quality data analysisreport of a wastewater treatment plant. Where the actual testing dataabout dissolved oxygen concentration DO, total nitrogen TN, loadactivated sludge F/M, power of hydrogen pH and temperature T is selectedfor the experimental sample data. There are 1000 groups data areavailable after eliminate the abnormal, where 500 group used as trainingsamples and the rest of 500 as test sample. The technical scheme andimplementation steps as following.

An intelligent identification method for sludge bulking based on atype-2 fuzzy-neural-network comprise the following steps:

(1) Determine the input and output variables of sludge volume index(SVI): in the activated sludge wastewater treatment process, the inputvariables of SVI soft-computing model include: dissolved oxygen (DO)concentration, total nitrogen (TN) concentration, organic load rate(F/M), pH, T. The output value of soft-computing model is the SVIvalues. The sludge bulking contains the following fault types: low DOconcentration, nutrient deficit, low sludge loading, low pH, and lowtemperature;

(2) SVI soft-computing model: establish SVI soft-computing model basedon type-2 fuzzy-neural-network, the structure of type-2fuzzy-neural-network contains five layers: input layer, membershipfunction layer, firing layer, consequent layer and output layer, thenetwork is 5-15-3-2-1, including 5 neurons in input layer, 15 neurons inmembership function layer, 3 neurons in firing layer, 2 neurons inconsequent layer and 1 neurons in output layer; connecting weightsbetween input layer and membership function layer are assigned 1; thenumber of the training sample is N, the input of type-2fuzzy-neural-network is x(t)=[x₁(t), x₂(t), x₃(t), x₄(t), x₅(t)] at timet, x₁(t) represents the DO concentration at time t; x₂(t) represents theTN concentration at time t, x₃(t) represents the organic load rate (F/M)value at time t, x₄(t) represents the pH value at time t, and x₅(t)represents the T value at time t, the output of type-2fuzzy-neural-network is y_(d)(t) and the actual output is y(t); type-2fuzzy-neural-network includes:

{circle around (1)} input layer: there are 5 neurons in this layer, theoutput is:o _(i)(t)=x _(i)(t)  (1)where o_(i)(t) is the ith output value at time t, i=1, 2, . . . , 5,

{circle around (2)} membership function layer: there are M neurons inmembership function layer, the output is:

$\begin{matrix}{{{\tau_{m}^{i}(t)} = {{N\left( {{c_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)} = {\exp\left\{ {{- \frac{1}{2}}\left( \frac{{o_{i}(t)} - {c_{m}^{i}(t)}}{\sigma_{m}^{i}(t)} \right)^{2}} \right\}}}},{i = 1},2,\ldots\;,{5;{m = 1}},2,\ldots\;,M,} & (2) \\{{{c_{m}^{i}(t)} \in \left\lbrack {{{\underset{\_}{c}}_{m}^{i}(t)},{{\overset{\_}{c}}_{m}^{i}(t)}} \right\rbrack},} & (3)\end{matrix}$where τ^(i) _(m)(t) is the mth membership function with the ith input attime t, N represents the membership function, c^(i) _(m)(t) is theuncertain center of the mth membership function neuron with the ithinput at time t, c^(i) _(m)(t) is the lower center value of the mthmembership function neuron with the ith input at time t, c ^(i) _(m)(t)is the upper center value of the mth membership function neuron with theith input at time t (where the initial lower center value and initialupper center value of the mth membership function neuron with the ithinput i.e., c ^(i) _(m)(0) and c ^(i) _(m)(t) is obtained by that randominitial center of the mth membership function neuron with the ith inputc^(i) _(m)(0) add and subtract a constant), σ^(i) _(m)(t) is thestandard deviation of the mth membership function neuron with the ithinput at time t, the bounded internal of τ^(i) _(m)(t) is [τ ^(i)_(m)(t), τ ^(i) _(m)(t)]

$\begin{matrix}{{{\underset{\_}{\tau}}_{m}^{i}\left( {o_{i}(t)} \right)} = \left\{ {\begin{matrix}{{N\left( {{{\overset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} \leq {\left( {{{\underset{\_}{c}}_{m}^{i}(t)} + {{\overset{\_}{c}}_{m}^{i}(t)}} \right)/2}} \\{{N\left( {{{\underset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} > {\left( {{{\underset{\_}{c}}_{m}^{i}(t)} + {{\overset{\_}{c}}_{m}^{i}(t)}} \right)/2}}\end{matrix},} \right.} & (4) \\{{{\overset{\_}{\tau}}_{m}^{i}\left( {o_{i}(t)} \right)} = \left\{ {\begin{matrix}{{N\left( {{{\underset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} \leq {{\underset{\_}{c}}_{m}^{i}(t)}} \\{1,} & {{{\underset{\_}{c}}_{m}^{i}(t)} < {o_{i}(t)} < {{\overset{\_}{c}}_{m}^{i}(t)}} \\{{N\left( {{{\overset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},} & {{o_{i}(t)} > {{\overset{\_}{c}}_{m}^{i}(t)}}\end{matrix},} \right.} & (5)\end{matrix}$where τ ^(i) _(m)(t) and τ ^(i) _(m)(t) are the lower value and uppervalue of the mth membership function neuron with the ith input at timet, respectively, 0<τ ^(i) _(m)(t)≤τ ^(i) _(m)(t)≤1,

{circle around (3)} firing layer: there are L neurons in this layer, andthe output values are:

$\begin{matrix}{{{F_{l}(t)} = \left\lbrack {{{\underset{\_}{f}}_{l\;}(t)},{{\overset{\_}{f}}_{l}(t)}} \right\rbrack},{{{\underset{\_}{f}}_{l\;}(t)} = {\prod\limits_{i = 1}^{5}{{\underset{\_}{\tau}}_{m}^{i}(t)}}},{{{\overset{\_}{f}}_{l}(t)} = {\prod\limits_{i = 1}^{5}{{\overset{\_}{\tau}}_{m}^{i}(t)}}},{l = 1},2,\ldots\mspace{14mu},L,} & (6)\end{matrix}$where F_(l)(t) is the firing strength of the lth firing neuron, f_(l)(t) and f _(l)(t) are the lower firing strength and upper firingstrength of the lth firing neuron, respectively, 0<f _(l)t)≤f _(l)(t)≤1,

{circle around (4)} consequent layer: this layer contains two consequentneurons, the output values are

$\begin{matrix}{{{\underset{\_}{y}(t)} = \frac{\sum\limits_{l = 1}^{L}{{{\underset{\_}{f}}_{l}(t)}{a_{l}(t)}}}{\sum\limits_{l = 1}^{L}{{\underset{\_}{f}}_{l}(t)}}},\mspace{14mu}{{\overset{\_}{y}(t)} = \frac{\sum\limits_{l = 1}^{L}{{{\underset{\_}{f}}_{l}(t)}{a_{l}(t)}}}{\;{\sum\limits_{l = 1}^{L}{{\underset{\_}{f}}_{l}(t)}}}},} & (7) \\{{{a_{l}(t)} = {\sum\limits_{l = 1}^{5}{{\theta_{l}^{i}(t)}{x_{i}(t)}}}},} & \;\end{matrix}$where y(t) and y(t) are the low and up output values of the consequentneurons, a_(i)(t) is weight of the lth firing neuron, θ^(i) _(l)(t) isthe weight coefficient of the lth firing neuron with the ith input attime t,

{circle around (5)} output layer: the output value is:y(t)=η(t) y (t)+(1−η(t)) y (t)  (8)where η(t) and y(t) are the proportion of the low output and the outputvalue of type-2 fuzzy-neural-network, the error of type-2fuzzy-neural-network is:

$\begin{matrix}{{E(t)} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {{y_{d}(t)} - {y(t)}} \right)^{2}}}} & (9)\end{matrix}$where y_(d)(t) is the output of type-2 fuzzy-neural-network and theactual output is expressed as y(t);

(3) train type-2 fuzzy-neural-network

{circle around (1)} give the type-2 fuzzy-neural-network, the initialnumber of firing layer neurons is M, M>2 is a positive integer; theinput of type-2 fuzzy-neural-network is x(1), x(2), . . . , x(t), . . ., x(N), correspondingly, the output is y_(d)(1), y_(d)(2), . . . ,y_(d)(t), . . . , y_(d)(N), expected error value is set to E_(d),E_(d)∈(0, 0.01),

{circle around (2)} set the learning step s=1;

{circle around (3)} t=s; according to Eqs. (1)-(7), calculate the outputof type-2 fuzzy-neural-network, exploiting adaptive second-orderalgorithm:ψ(t+1)=ψ(t)+(H(t)+λ(t)I)⁻¹ v(t)  (10)where ψ(t)=[c ^(i) _(m)(t), c^(i) _(m)(t), σ^(i) _(m)(t), η(t), w^(i)_(m)(t)] is the parameter matrix of type-2 fuzzy-neural-network at timet, c^(i) _(m)(t) is the lower center value of the mth membershipfunction neuron with the ith input at time t, c ^(i) _(m)(t) is theupper center value of the membership function neuron with the ith inputat time t, σ^(i) _(m)(t) is the standard deviation of the mth membershipfunction neuron with the ith input at time t, η(t) is the proportion ofthe lower output, θ^(i) _(l)(t) is the weight coefficient of the lthfiring neuron with the ith input at time t, H(t) is the quasi Hessianmatrix, v(t) is gradient vector, I is the identity matrix and λ(t) isthe adaptive learning rate defined as:λ(t)=γ|E(t)|+(1−γ)∥v(t)∥  (11)where γ∈(0, 1), the expression of H(t) and v(t) are defined as:H(t)=J ^(T)(t)J(t)  (12)v(t)=J ^(T)(t)E(t)  (13)where the Jacobian vector J(t) is calculated as:

$\begin{matrix}{{J(t)} = \left\lbrack {\frac{\partial{e(t)}}{\partial{{\underset{\_}{c}}_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{{\overset{\_}{c}}_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{\sigma_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{\eta(t)}},\frac{\partial{e(t)}}{\partial{\theta_{m}^{i}(t)}}} \right\rbrack} & (14)\end{matrix}$

{circle around (4)} according to Eq. (9), calculate the performance oftype-2 fuzzy-neural-network, if E(t)≥E_(d), go to step {circle around(3)}; if E(t)<E_(d), stop the training process;

The predicted results of sludge volume index SVI is shown in FIG. 2, theX axis: sample, the unit is a; Y axis: the test output of sludge volumeindex SVI, the unit is ml/g; the blue line is the predict output valueof sludge volume index SVI; the black line is actual output values ofsludge volume index SVI; The error between the predicted output and theactual output of sludge volume index (SVI) is shown in FIG. 3. The Xaxis: sample, the unit is a; Y axis: the test output of sludge volumeindex SVI, the unit is ml/g;

(4) The target-related identification algorithm is used to determine thefault type of sludge bulking, which is specifically as follows:

{circle around (1)} the test samples is used as the input of the type-2fuzzy-neural-network, and the sludge volume index (SVI) is calculated;

{circle around (2)} if SVI≤150, it is determined that there is no sludgebulking during the wastewater treatment process;

{circle around (3)} if SVI>150, sludge bulking at the wastewatertreatment operation was determined and regression coefficients of allvariables were calculated:

$\begin{matrix}{{{b_{i}(t)} = \frac{{u_{i}(t)}^{T}{t_{i}(t)}}{{t_{i}(t)}^{T}{t_{i}(t)}}},} & (15)\end{matrix}$where b_(i)(t) is the regression coefficient of ith input at time t,b(t)=[b₁(t), . . . , b_(i)(t), . . . , b₅(t)] is the regressioncoefficient vector, u_(i)(t) is the ith score vector of the outputvector at time t, U(t)=[u₁(t), . . . . , u_(i)(t), . . . , u₅(t)] isscore matrix of the output vector at time t, t_(i)(t) is the ith scorevector of the input matrix at time t, T(t)=[t_(i)(t), . . . , t_(i)(t),. . . , t₅(t)] is score matrix of the input matrix at time t, u_(i)(t)and t_(i)(t) are given as

$\begin{matrix}{{{u_{i}(t)} = \frac{{y(t)}{q_{i}(t)}}{{q_{i}(t)}^{T}{q_{i}(t)}}},} & (16) \\{{{t_{i}(t)} = \frac{{X(t)}{w_{i}(t)}}{{w_{i}(t)}^{T}{w_{i}(t)}}},} & (17)\end{matrix}$where q_(i)(t) is the ith loading value of output vector at time t,q(t)∈R^(1×5) is the loading vector of output vector at time t,y(t)=[y(t−K+1), y(t−K+2), . . . , y(t)]^(T), y(t) is the SVI value attime t, X(t)=[x₁(t), . . . , x_(i)(t), . . . , x₅(t)] is the inputmatrix of type-2 fuzzy-neural-network, x_(i)(t)=[x_(i)(t−K+1),x_(i)(t−K+2), . . . , x_(i)(t)]^(T), x_(i)(t) is the ith input variableat time t, w_(i)(t) is the ith feature vector at time t, W(t)=[w₁(t), .. . , w_(i)(t), . . . , w₅(t)] is the feature matrix of X(t)^(T)y(t),the expressions of q_(i)(t) and W(t) are

$\begin{matrix}{{{q_{i}(t)}^{T} = \frac{{t_{i}(t)}^{T}{y(t)}}{{t_{i}(t)}^{T}{t_{i}(t)}}},} & (18) \\{{{{W(t)}^{T}{\Lambda(t)}{W(t)}} = {E\left\{ {{X(t)}^{T}{y(t)}{y(t)}^{T}{X(t)}} \right\}}},} & (19)\end{matrix}$where Λ(t) is the eigenvalue matrices of X(t)^(T)y(t). The function Erepresents the eigenvector and eigenvalue of the matrix, the innerrelative model of y(t) and X(t) can be expressed as:

$\begin{matrix}\left\{ {\begin{matrix}{{X(t)} = {{{{T(t)}{P(t)}^{T}} + {\Delta(t)}} = {{\sum\limits_{i = 1}^{5}{{t_{i}(t)}{p_{i}(t)}^{T}}} + {\Delta(t)}}}} \\{{y(t)} = {{{{U(t)}{q(t)}^{T}} + {G(t)}} = {{\sum\limits_{i = 1}^{5}{{u_{i}(t)}{q_{i}(t)}^{T}}} + {G(t)}}}}\end{matrix},} \right. & (20)\end{matrix}$where Δ(t)∈R^(K×5) is the residual matrix of X(t), Δ(t)=[δ₁(t), . . . ,δ_(i)(t), . . . , δ₅(t)], where δ_(i)(t) present the residual vector ofith input. G(t)∈R^(K×1) is the residual vector of y(t);

{circle around (4)} when the regression coefficient of the inputvariable satisfies:b _(max)(t)=max b(t),  (21)where b_(max)(t) is the maximum regression coefficient of the inputvariables, and the corresponding fault type is the source of sludgebulking.

FIG. 4 shows the fault classification of sludge bulking. X-axis: inputvariable, no unit, Y-axis: contribution, no unit.

What is claimed is:
 1. An intelligent identification method for sludgebulking based on a type-2 fuzzy-neural-network, comprising the followingsteps: (1) determine input and output variables of sludge volume index(SVI): in an activated sludge wastewater treatment process, the inputvariables of SVI soft-computing model include: dissolved oxygen (DO)concentration, total nitrogen (TN) concentration, organic load rate(F/M), pH, T, output values of the soft-computing model are SVI values,the sludge bulking contains the following fault types: low DOconcentration, nutrient deficit, low sludge loading, low pH, and lowtemperature; (2) SVI soft-computing model: establish the SVIsoft-computing model based on type-2 fuzzy-neural-network, a structureof type-2 fuzzy-neural-network contains five layers: input layer,membership function layer, firing layer, consequent layer and outputlayer, the network is 5-M-L-2-1, including 5 neurons in the input layer,M neurons in the membership function layer, L neurons in the firinglayer, 2 neurons in the consequent layer and 1 neurons in the outputlayer, M and L are integers larger than 2; connecting weights betweenthe input layer and the membership function layer are assigned 1; thenumber of training samples is N, the input of type-2fuzzy-neural-network is x(t)=[x₁(t), x₂(t), x₃(t), x₄(t), x₅(t)] at timet, x₁(t) represents DO concentration at time t; x₂(t) represents TNconcentration at time t, x₃(t) represents an organic load rate (F/M)value at time t, x₄(t) represents pH value at time t, and x₅(t)represents T value at time t, the output of type-2 fuzzy-neural-networkis y_(d)(t) and an actual output is y(t); type-2 fuzzy-neural-networkincludes: an input layer: there are 5 neurons in this layer, the outputis:o _(i)(t)=x _(i)(t)  (1) where o_(i)(t) is the ith output value at timet, i=1, 2, . . . , 5, a membership function layer: there are M neuronsin the membership function layer, the output is: $\begin{matrix}{{{\tau_{m}^{i}(t)} = {{N\left( {{c_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)} = {\exp\left\{ {{- \frac{1}{2}}\left( \frac{{o_{i}(t)} - {c_{m}^{i}(t)}}{\sigma_{m}^{i}(t)} \right)^{2}} \right\}}}},{i = 1},2,\ldots\mspace{14mu},{5;{m = 1}},2,\ldots\mspace{14mu},M,} & (2) \\{{{c_{m}^{i}(t)} \in \left\lbrack {{{\underset{\_}{c}}_{m}^{i}(t)},{{\overset{¯}{c}}_{m}^{i}(t)}} \right\rbrack},} & (3)\end{matrix}$ where τ^(i) _(m)(t) is the mth membership function withthe ith input at time t, N represents the membership function, c^(i)_(m)(t) is the uncertain center of the mth membership function neuronwith the ith input at time t, c^(i) _(m)(t) is the lower center value ofthe mth membership function neuron with the ith input at time t, c ^(i)_(m)(t) is the upper center value of the mth membership function neuronwith the ith input at time t (where the initial lower center value andinitial upper center value of the mth membership function neuron withthe ith input, i.e., c ^(i) _(m)(0) and c ^(i) _(m)(t) is obtained bythat random initial center of the mth membership function neuron withthe ith input c^(i) _(m)(0) add and subtract a constant), σ^(i) _(m)(t)is the standard deviation of the mth membership function neuron with theith input at time t, the bounded internal of τ^(i) _(m)(t) is [τ ^(i)_(m)(t), τ ^(i) _(m)(t)] $\begin{matrix}{{{\underset{\_}{\tau}}_{m}^{i}\left( {o_{i}(t)} \right)} = \left\{ {\begin{matrix}{{N\left( {{{\overset{¯}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},{{o_{i}(t)} \leq {\left( {{{\underset{\_}{c}}_{m}^{i}(t)} + {{\overset{¯}{c}}_{m}^{i}(t)}} \right)/2}}} \\{{N\left( {{{\underset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},{{o_{i}(t)} > {\left( {{{\underset{\_}{c}}_{m}^{i}(t)} + {{\overset{¯}{c}}_{m}^{i}(t)}} \right)/2}}}\end{matrix},} \right.} & (4) \\{{{\overset{¯}{\tau}}_{m}^{i}\left( {o,(t)} \right)} = \left\{ {\begin{matrix}{{N\left( {{{\underset{\_}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},{{o_{i}(t)} \leq {{\underset{¯}{c}}_{m}^{i}(t)}}} \\\begin{matrix}{1,} & {{{\underset{¯}{c}}_{m}^{i}(t)} < {o_{i}(t)} < {{\overset{¯}{c}}_{m}^{i}(t)}}\end{matrix} \\{{N\left( {{{\overset{¯}{c}}_{m}^{i}(t)},{{\sigma_{m}^{i}(t)};{o_{i}(t)}}} \right)},{{o_{i}(t)} > {{\overset{¯}{c}}_{m}^{i}(t)}}}\end{matrix},} \right.} & (5)\end{matrix}$ where τ ^(i) _(m)(t) and τ ^(i) _(m)(t) are the lowervalue and upper value of the mth membership function neuron with the ithinput at time t, respectively, 0<τ ^(i) _(m)(t)≤τ ^(i) _(m)(t)≤1, afiring layer: there are L neurons in this layer, and the output valuesare: $\begin{matrix}{{{F_{l}(t)} = \left\lbrack {{{\underset{¯}{f}}_{l}(t)},{{\overset{¯}{f}}_{l}(t)}} \right\rbrack},{{{\underset{¯}{f}}_{l}(t)} = {\prod\limits_{i = 1}^{5}{{\underset{¯}{\tau}}_{m}^{i}(t)}}},{{{\overset{¯}{f}}_{l}(t)} = {\prod\limits_{i = 1}^{5}{{\overset{¯}{\tau}}_{m}^{i}(t)}}},{l = 1},2,\ldots\mspace{14mu},L,} & (6)\end{matrix}$ where F_(l)(t) is the firing strength of the lth firingneuron, f _(l)(t) and f _(l)(t) are the lower firing strength and upperfiring strength of the lth firing neuron, respectively, 0<f _(l)t)≤f_(l)(t)≤1, a consequent layer: this layer contains two consequentneurons, the output values are $\begin{matrix}{{{\underset{¯}{y}(t)} = \frac{\sum\limits_{l = 1}^{L}{{{\underset{¯}{f}}_{l}(t)}{a_{l}(t)}}}{\sum\limits_{l = 1}^{L}{{\underset{¯}{f}}_{l}(t)}}},{{\overset{¯}{y}(t)} = \frac{\sum\limits_{l = 1}^{L}{{{\underset{¯}{f}}_{l}(t)}{a_{l}(t)}}}{\sum\limits_{l = 1}^{L}{{\underset{¯}{f}}_{l}(t)}}},{{a_{l}(t)} = {\sum\limits_{i = 1}^{5}{{\theta_{l}^{i}(t)}{x_{i}(t)}}}},} & (7)\end{matrix}$ where y(t) and y(t) are the low and up output values ofthe consequent neurons, a_(l)(t) is weight of the lth firing neuron,θ^(i) _(l)(t) is the weight coefficient of the lth firing neuron withthe ith input at time t, an output layer: the output value is:y(t)=η(t) y (t)+(1−η(t)) y (t)  (8) where η(t) and y(t) are theproportion of the low output and the output value of type-2fuzzy-neural-network, the error of type-2 fuzzy-neural-network is:$\begin{matrix}{{E(t)} = {\frac{1}{N}{\sum\limits_{t = 1}^{N}\left( {{y_{d}(t)} - {y(t)}} \right)^{2}}}} & (9)\end{matrix}$ where y_(d)(t) is the output of type-2fuzzy-neural-network and the actual output is expressed as y(t); (3)train type-2 fuzzy-neural-network (a) give the type-2fuzzy-neural-network, the initial number of firing layer neurons is M,M>2 is a positive integer; the input of type-2 fuzzy-neural-network isx(1), x(2), . . . , x(t), . . . , x(N), correspondingly, the output isy_(d)(1), y_(d)(2), . . . , y_(d)(t), . . . , y_(d)(N), expected errorvalue is set to E_(d), E_(d)∈(0, 0.01), (b) set the learning step s=1;(c) t=s; according to Eqs. (1)-(7), calculate the output of type-2fuzzy-neural-network, exploiting adaptive second-order algorithm:ψ(t+1)=ψ(t)+(H(t)+λ(t)I)⁻¹ v(t)  (10) where ψ(t)=[c ^(i) _(m)(t), c^(i)_(m)(t), σ^(i) _(m)(t), η(t), w^(i) _(m)(t)] is the parameter matrix oftype-2 fuzzy-neural-network at time t, c^(i) _(m)(t) is the lower centervalue of the mth membership function neuron with the ith input at timet, c ^(i) _(m)(t) is the upper center value of the membership functionneuron with the ith input at time t, σ^(i) _(m)(t) is the standarddeviation of the mth membership function neuron with the ith input attime t, η(t) is the proportion of the lower output, θ^(i) _(l)(t) is theweight coefficient of the lth firing neuron with the ith input at timet, H(t) is the quasi Hessian matrix, v(t) is gradient vector, I is theidentity matrix and λ(t) is the adaptive learning rate defined as:λ(t)=γ|E(t)|+(1−γ)∥v(t)∥  (11) where γ∈(0, 1), the expression of H(t)and v(t) are defined as:H(t)=J ^(T)(t)J(t)  (12)v(t)=J ^(T)(t)E(t)  (13) where the Jacobian vector J(t) is calculatedas: $\begin{matrix}{{J(t)} = \left\lbrack {\frac{\partial{e(t)}}{\partial{{\underset{¯}{c}}_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{{\overset{\_}{c}}_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{\sigma_{m}^{i}(t)}},\frac{\partial{e(t)}}{\partial{\eta(t)}},\frac{\partial{e(t)}}{\partial{\theta_{m}^{i}(t)}}} \right\rbrack} & (14)\end{matrix}$ (d) according to Eq. (9), calculate the performance oftype-2 fuzzy-neural-network, if E(t)≥E_(d), go to step (c); ifE(t)<E_(d), stop the training process; (4) the target-relatedidentification algorithm is used to determine the fault type of sludgebulking, which is specifically as follows: (a) the test samples are usedas the input of the type-2 fuzzy-neural-network, and the sludge volumeindex (SVI) is calculated; (b) if SVI≤150, it is determined that thereis no sludge bulking during the wastewater treatment process; (c) ifSVI>150, sludge bulking at the wastewater treatment operation isdetermined and regression coefficients of all variables are calculated:$\begin{matrix}{{{b_{i}(t)} = \frac{{u_{i}(t)}^{T}{t_{i}(t)}}{{t_{i}(t)}^{T}{t_{i}(t)}}},} & (15)\end{matrix}$ where b_(i)(t) is the regression coefficient of ith inputat time t, b(t)=[b₁(t), . . . , b_(i)(t), . . . , b₅(t)] is theregression coefficient vector, u_(i)(t) is the ith score vector of theoutput vector at time t, U(t)=[u₁(t), . . . , u_(i)(t), . . . , u₅(t)]is score matrix of the output vector at time t, t_(i)(t) is the ithscore vector of the input matrix at time t, T(t)=[t₁(t), . . . ,t_(i)(t), . . . , t₅(t)] is score matrix of the input matrix at time t,u_(i)(t) and t_(i)(t) are given as: $\begin{matrix}{{{u_{i}(t)} = \frac{{y(t)}{q_{i}(t)}}{{q_{i}(t)}^{T}{q_{i}(t)}}},} & (16) \\{{{t_{i}(t)} = \frac{{X(t)}{w_{i}(t)}}{{w_{i}(t)}^{T}{w_{i}(t)}}},} & (17)\end{matrix}$ where q_(i)(t) is the ith loading value of output vectorat time t, q(t)∈R^(1×5) is the loading vector of output vector at timet, y(t)=[y(t−K+1), y(t−K+2), . . . , y(t)]^(T), y(t) is the SVI value attime t, X(t)=[x₁(t), . . . , x_(i)(t), . . . , x₅(t)] is the inputmatrix of type-2 fuzzy-neural-network, x_(i)(t)=[x_(i)(t−K+1),x_(i)(t−K+2), . . . , x_(i)(t)]^(T), x_(i)(t) is the ith input variableat time t, w_(i)(t) is the ith feature vector at time t, W(t)=[w₁(t), .. . , w_(i)(t), . . . , w₅(t)] is the feature matrix of X(t)^(T)y(t),the expressions of q_(i)(t) and W(t) are $\begin{matrix}{{{q_{i}(t)}^{T} = \frac{{t_{i}(t)}^{T}{y(t)}}{{t_{i}(t)}^{T}{t_{i}(t)}}},} & (18) \\{{{{W(t)}^{T}{\Lambda(t)}{W(t)}} = {E\left\{ {{X(t)}^{T}{y(t)}{y(t)}^{T}{X(t)}} \right\}}},} & (19)\end{matrix}$ where Λ(t) is the eigenvalue matrices of X(t)^(T)y(t), thefunction E represents the eigenvector and eigenvalue of the matrix, theinner relative model of y(t) and X(t) can be expressed as:$\begin{matrix}{\left\{ \begin{matrix}{{X(t)} = {{{{T(t)}{P(t)}^{T}} + {\Delta(t)}} = {{\underset{i = 1}{\sum\limits^{5}}{{t_{i}(t)}{p_{i}(t)}^{T}}} + {\Delta(t)}}}} \\{{y(t)} = {{{{U(t)}{q(t)}^{T}} + {G(t)}} = {{\overset{5}{\sum\limits_{i = 1}}{{u_{i}(t)}{q_{i}(t)}^{T}}} + {G(t)}}}}\end{matrix} \right.,} & (20)\end{matrix}$ where Δ(t)∈R^(K×5) is the residual matrix of X(t),Δ(t)=[δ₁(t), . . . , δ_(i)(t), . . . , δ₅(t)], where δ_(i)(t) presentthe residual vector of ith input, G(t)∈R^(K×1) is the residual vector ofy(t); (d) when the regression coefficient of the input variablesatisfies:b _(max)(t)=max b(t),  (21) where b_(max)(t) is the maximum regressioncoefficient of the input variables, and the corresponding fault type isthe source of sludge bulking.